Manuscript XX-D@1 · XX-D.tex · section · c83a526d

A compact source cutoff, not a native replacement

The Hessian: microcausality, noncompactness, and internal contraction › A compact source cutoff, not a native replacement · lines 919–943

XX-D@1#eq:RRcut

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Choose $χ \in C_{c}^{\infty} (R^{3})$ , equal to one near the origin, and set

χ_{R} (y) = χ (y / R) .

Define

R_{R} z = - G_{sp} \partial_{i} (χ_{R} (\calCz)^{ia}) .

The derivative includes the cutoff derivative. Equivalently, (eq:P) has source $χ_{R} \calCz$ . Let $F_{f, R}$ and $K_{f, R}$ replace $R$ by $R_{R}$ .

This is a compact source approximation. It is not the native rod prescription at finite $R$ , and it is not ultraviolet smoothing: the equal-time delta and Newton singularity are unchanged.

Displayed formulas in this passage

XX-D@1#eq:RRcut · line 927
$R_{R} z = - G_{sp} \partial_{i} (χ_{R} (\calCz)^{ia}) .$
unlabeled · line 923
$χ_{R} (y) = χ (y / R) .$